The reduction in focal volume caused by harmonics in the hologram field is useful.
For example, it allows for small foci to be generated without the need to pulse or design the hologram for a high frequency. However, the generation of additional foci is undesirable. For example, in therapeutic ultrasound it could result in heating at targets away from the desired focus. For other applications, the additional foci are less of a concern (i.e., in imaging they could be windowed out). However, adjusting the design of the zone plate to account for the presence of these harmonics is still worthwhile. This can be done by altering the thickness of each ring on the zone plate to change the distribution of pressure generated by each harmonic at its design focus and at each of its higher-order foci. As the zone plate is designed only to optimise focusing for the fundamental, when harmonics are present, this is not necessarily optimal.
For a zone plate designed to focus a frequency f at a distance d away from the zone plate, the relative maximum pressure pmax( f ) generated by the fundamental and the first two harmonics at the focus d can be approximated by the following equations
pmax( f ) ∝ Z π
0
sin(θ )dθ , (3.12)
pmax( f2) ∝ 2 Z 2π
0
sin(θ )dθ , (3.13)
pmax( f3) ∝ 3 Z 3π
0
sin(θ )dθ . (3.14)
Here pmax( f2) and pmax( f3) are the maximum pressures of the second and third harmonic respectively, and the factors of 2 and 3 in Eq. 3.13 and 3.14 are added
to account for the scaling of focal gain with frequency. These equations derive from the path length difference between acoustic waves arriving from the inner and outer radii of a ring being λ /2, which corresponds to a phase difference of π for the fundamental. It can be seen from Eq. 3.13 that the second harmonic almost entirely cancels out at the focus, and from Eq. 3.12 & 3.14 that the third harmonic and fundamental should generate similar pressures.
Alternatively, consider a zone plate calculated for a frequency of f3 to focus at a distance d away from the zone plate from which 2 out of 3 rings have been removed. This has an equivalent effect to adjusting the thickness of each of the rings on the fundamental zone plate by a specific amount. An example is shown in Fig. 3.5 (a). For this zone plate the maximum pressure pmax( f ) generated by the fundamental and the first two harmonics relative to each other can be approximated by
pmax( f ) ∝ Z 2π3
π 3
sin(θ )dθ , (3.15)
pmax( f2) ∝ 2 Z 5π
6 π 6
sin(θ )dθ , (3.16)
pmax( f3) ∝ 3 Z π
0
sin(θ )dθ . (3.17)
This was derived from the observation that the path length difference between acoustic waves arriving from the inner and outer radii of each ring is 0.5λ3 . This is known because the harmonic zone plate was calculated initially from a frequency of f3to focus at an identical point to the original zone plate. So geometrically, the path length difference for each ring remaining on the zone plate must be 0.5λ3 . This corresponds to a phase difference of π3 for f , π2 for f2 and π for f3. Comparing Eq. 3.12-3.14 and 3.15-3.17, the relative maximum pressure predicted by the two is 1:1.19. This is because, the second harmonic now contributes to the pressure generated at the target focus, and the fundamental and third harmonic have not sig-nificantly reduced. This suggests that despite having lower number of transmitting
pixels, the harmonic zone plate should generate higher peak pressures. Extending this to 5 harmonics, the predicted ratio becomes 1:1.24.
A simulation was carried out to validate these approximations. The field from Fig. 3.3 (c), which was generated by a 5 MHz zone plate with a 1.5×1.5 cm aperture and a focus at 1.5 cm, was compared against the field generated by a harmonic zone plate simulated on the same grid for the same input (i.e., same number of harmonics). The harmonic zone plate used is shown in Fig. 3.5 (a). This was generated by first calculating a hologram for a 15 MHz design frequency with a 1.5×1.5 cm aperture and a focus at 1.5 cm, then removing two out of three rings.
Maximum amplitude projections of both acoustic fields are shown in Fig. 3.5 (b)-(c).
It can be seen that the maximum pressure generated by the harmonic zone plate is greater than the conventional zone plate. The ratio between the two differs slightly from the prediction due to the maxima generated for different harmonics not temporally coinciding due to phase differences, and because the input spectra in the simulation wasn’t perfectly flat. Other changes between the two fields are that the volume of the focus generated by the harmonic zone plate is decreased by a further 33% relative to the conventional zone plate, the signal to background ratio is improved, and the pressure of the additional foci is decreased. The spectra at the design focus was measured in both simulations, these are shown in Fig. 3.5 (d) and (e). These show that, as predicted, the second and fourth harmonics are almost completely absent for the conventional zone plate and for the harmonic zone plate they have a larger amplitude.
These results suggests that harmonic zone plates could outperform conven-tional zone plates for broadband pulsed input sources. The lower number of trans-mitting pixels on the harmonic zone plate also means that for a hologram generated by spatial modulation of an incident laser pulse, the optical fluence for each pixel could be greatly increased.
depth (mm)
Figure 3.5: (a) 15 MHz harmonic zone plate, designed for a 3×3 cm aperture with a focus at 3 cm. (b) Acoustic field from Fig. 3.3(a), pressure normalised relative to maximum in (c). (c) Normalised maximum amplitude projection of the max-imum pressure generated by the hologram from (a) with a pulsed input. The harmonic zone plate generates greater pressure and has lower background com-pared to the conventional zone plate. (d) Spectra generated at target focus for conventional zone plate. (e) Spectra generated at target focus for harmonic zone plate.